What does it mean to say that some fact or state of affairs is necessary? Well it is to say that things couldn’t possibly have been otherwise. But, as I have argued previously, there is more than one kind of possibility. So in what kind of possibility do we have in mind when we talk about necessity?
Well obviously there isn’t one right answer, it depends on the context. In the broadest sense of possibility, metalogical possibility, nothing is necessary, since everything is possible (even apparently contradictory states of affairs). Obviously this isn’t useful in the least, so we can ignore it from this point forwards. More usefully we have logical possibility. Under logical possibility only logical truths (truths that can be proved using only the logical axioms) are necessary. If we understand logical truths to be the a priori facts (things which can be known purely by logical deduction, without the aid of experience) then everything that is necessary is also a priori.
Obviously this contradicts claims made by Kripke about possibility, as he claimed that some facts were necessarily true a posteriori. But, as shown previously, if we understand a possible world as an arbitrary logical model, then Kipke’s claim about the necessity of identity can in fact shown to be false (meaning that his proof that identity statements are necessary is in fact in error). This, combined with the fact that everything necessary in this wide sense of possibility is a logical truth, leads us to the conclusion that everything necessary can be known a priori, and vice versa.
But there is a third kind of possibility, restricted possibility. Because in restricted possibility certain background conditions are held constant facts besides logical truths may be necessary. Specifically, the background conditions, plus everything that logically is entailed by them, is necessary. This actually allows us to recover the claim that identity is necessary, if the possibilities are restricted in the right way. One such restriction that will work is if we consider only the possible rearrangements of individuals (meaning that different relationships and properties may hold of them, but that the set of individuals itself is held constant). And under this restriction the identity of individuals is not only necessary, but a posteriori (since even knowing that the set of individuals is a constant doesn’t tell us which individuals are identical). But, on the other hand, I can’t see how this particular restriction is a useful tool to work with (under what situations are only the rearrangements of individuals possibilities?).
On the other hand, restricting possibility to the physical possibilities (meaning universes with different initial states but the same physical laws) is much more useful. In such a situation the necessary facts reveal certain identity or supervenience relations. For example, it is a necessary fact that molecular motion causes phase changes, combustion, ect. And thus heat (the cause of those things) simply is molecular motion (or supervenes on molecular motion). Of course such truths are still a posteriori, since we don’t know the physical laws conceptually, and thus can’t say exactly what is and what isn’t a possibility without going out and discovering empirically these limitations.
An important question to ask here is that even given this disambiguation is necessity a useful tool? One way necessity is standard used is to formally capture certain kinds of causation. For example, “if X is a sphere then X appears round” is a perfectly valid conditional. But if X is not a sphere then “if X is a sphere then X appears square” is also perfectly valid. The first is, supposedly, telling us something more, so we instead say that “necessarily, if X is a sphere then X appears round”. And this indicates that there is a connection between the X being spherical and X looking round, even if X is not a sphere at the moment. But, when we put this in terms of restricted possibility, all we are really saying is that “if X is a sphere then X appears round” is a background restriction. (If we were in first order logic we could indicate that by putting the statement into our set of premises.) But, given that, we can say that there is no longer a need to claim that the fact is necessary, we have already captured all the relevant information.
Of course claims about necessity can be embedded into logical statements (for example “if necessarily φ then necessarily ψ”), and this does tell us something more. But, given that we can capture necessity in terms of restricted possibility, and given that we can capture restricted possibility by adding statements to our set of premises (from which we prove all our conclusions, the facts that are always true) it would seem like a better idea to simply provide a way to give this additional structure to the set of premises directly. The idea of necessity already blurs the line between the sentences in a formal language and the truth conditions of that language, so for clarity’s sake alone it would make sense to make these connections explicit.